Introverts aren't rude. Introverts are still polite and understand that they aren't going to be 100 percent "comfortable" all the time. Introverts can have perfectly lively conversations with their families. This isn't a personality trait. It's a behavorial thing. 

I have no advice for you. I'm sorry you are dealing with that. I have a 14 year old son who is incredibly immature for his age at times and at others very mature. I've been told that he is rude and disrespectful and I just haven't been able to get his head on straight. Hopefully, she comes around. 

what you mean by not wanting to be part of the family. I have two kids who are content to be on their own 90% of the time. They are a lot of fun when they are in the mood to be in a group but generally they are happier reading, creating, etc all by themselves. I spent most of my childhood and teen years reading, even at family events. Generally speaking I always let them be. I actually find my two who are very social to be more exhausting. 

Disrespect or being rude is different. Disrespect is not tolerated in my home, even once the kids are adults. And the kids always had responsbilities and chores, I just didn't do "forced family time" with my kids. 

A couple of questions:

1.) Why does she HAVE to live with you? To make you more comfortable? I've mentioned this to you before: you have allowed others to behave in such a way that makes YOU comfortable, even if it hurts other people. This sounds very similar.

2.) Why do you attribute her dismissiveness to her introvertedness? My brother is one step above recluse in his introvertedness. However, on the occasions that he is around family, he is polite, engages in conversation, participates in games, etc. Now, he can only do this for a few hours and only wants to a few times a year. We respect that for him and invite him to everything, realizing he likely won't show (and tell/show him it's okay to say no).

Being dismissive isn't a trait of being an introvert. That is a trait of being rude. There is a way of telling folks "thank you for engaging with me, but I am not comfortable/don't have the energy to reciprocate right now". Not wanting to participate isn't the problem; it's the attitude by which that message is delivered.

3.) Have you talked to your daughter about how to include her in a way that makes her comfortable? If everyone in the family is extroverted except her (and I'm assuming that she actually is introverted but hasn't been taught how to communicate effectively with others), then folks are going to have to include activities she will participate in (e.g. movie night, smaller groups, shorter activities) or be okay with her not participating when she POLITELY declines.

Sending your DD to her dad for two months wasn't going to magically make this all better. Your ex may do a better job of teaching her how to be an introvert in multi-vert world and living with him may be a better solution than living with you. If you're still struggling, I recommend parenting classes and counseling for both you and DD (individually and together) to figure out how to proceed with your relationship.

Wow just read through your previous posts are you sure your marriage is even salvageable at this point? If not let it go and focus on your daughter.

Personally if my SO's family trashed me and then they just forgave their family and carried on a relationship without me being involved that would be a nail in the coffin for me. 

Why don't you just live your own life and quit trying to control how they are and to each other? Do your own thing like your career and hobbies and friends?  And if they get in your face just say "huh? What's going on?" Why do you have to monitor introverted daughter and overly emotional husband? Just live your life. So daughter doesn't want to come over. Great. Husband is having a hissy fit because stepkids didn't say hello. So? Tell them you have a yoga class, sorry.

my SD used to throw fits all the time. I stopped spending time with her. My husbands cousin tried to start shit with me and my relationship. I blocked her. MIL tried to start gossip.  I said "huh? What are you talking about?" Let that crap roll off your back and focus on you. They will beg for your time.

It's a rough time with all the changes and hormones, add divorce dynamics and it's pretty confusing for kids.  I wish I could go back and just enjoy the person my daughter was as a teen rather than trying to mold her into what I wanted her to be.  She was my first, very rebellious to my conservative rules. She also chose to live with dad who let her do whatever and was a completely hands off parent.  

My advice is to remember that this time will pass quickly.  She will be an adult soon and will get to choose whether or not she wants a relationship with you. Now is the time to set the foundation of a good relationship for the future. Listen to her, whatever she wants to talk about. Be available when she needs you.  Even though she's acting tough and distant she still needs her mama.  

Why can't she stay with her father if she's happier there? Of course, she shouldn't be rude to anyone, but perhaps his home is quieter, with less drama, and that meets her needs?  As an introvert myself, I can't stand noise, drama, stress in my home - I need peace and quiet to thrive.

She's 16, let her stay where she wants to stay, and work on your relationship from afar.